FUNCTION: TableXX - table of all double Schubert polynomials

CALLING SEQUENCE:

TableXX(n)

TableXX(n, perm)

SP[TableXX](n)

SP[TableXX](n, perm)

PARAMETERS:

n = any positive integer denoting the degree of a symmetric group

perm = any list denoting a permutation

SYNOPSIS:

• The TableXX function returns the table of all double Schubert polynomials indexed by permutations in Sn. The polynomials are expressed in the basis of monomials.

• When the function is called with the second argument perm, the second alphabet (the yi's) is specialized as the permutation perm of the first one (the xi's): in other words, y.i is sent to x.perm[i].

• Whenever there is a conflict between the function name TableXX and another name used in the same session, use the long form SP['TableXX'].

EXAMPLES:

> with(SP):
> t:=TableXX(3);
 
             t := table([
                      [1, 2, 3] = 1
                      [3, 1, 2] = (x1 - y2) (x1 - y1)
                      [2, 1, 3] = x1 - y1
                      [1, 3, 2] = x2 - y2 + x1 - y1
                      [2, 3, 1] = (x2 - y1) (x1 - y1)
                      [3, 2, 1] = (x2 - y1) (x1 - y2) (x1 - y1)
                  ])
 
> t[ [3,1,2] ];
 
                            (x1 - y2) (x1 - y1)
 
> TableXX(3, [3,2,1]);
 
                 table([
                     [1, 2, 3] = 1
                     [3, 1, 2] = (x1 - x2) (x1 - x3)
                     [2, 1, 3] = x1 - x3
                     [1, 3, 2] = x1 - x3
                     [2, 3, 1] = (x2 - x3) (x1 - x3)
                     [3, 2, 1] = (x2 - x3) (x1 - x2) (x1 - x3)
                 ])